Theorem resco 5108

Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resco	|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Proof of Theorem resco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4912	. 2 |- Rel ( ( A o. B ) |` C )
2		relco 5102	. 2 |- Rel ( A o. ( B |` C ) )
3		vex 2729	. . . . . 6 |- x e. _V
4		vex 2729	. . . . . 6 |- y e. _V
5	3, 4	brco 4775	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
6	5	anbi1i 454	. . . 4 |- ( ( x ( A o. B ) y /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
7		19.41v 1890	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
8		an32 552	. . . . . 6 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
9		vex 2729	. . . . . . . 8 |- z e. _V
10	9	brres 4890	. . . . . . 7 |- ( x ( B |` C ) z <-> ( x B z /\ x e. C ) )
11	10	anbi1i 454	. . . . . 6 |- ( ( x ( B |` C ) z /\ z A y ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
12	8, 11	bitr4i 186	. . . . 5 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( x ( B |` C ) z /\ z A y ) )
13	12	exbii 1593	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
14	6, 7, 13	3bitr2i 207	. . 3 |- ( ( x ( A o. B ) y /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
15	4	brres 4890	. . 3 |- ( x ( ( A o. B ) |` C ) y <-> ( x ( A o. B ) y /\ x e. C ) )
16	3, 4	brco 4775	. . 3 |- ( x ( A o. ( B |` C ) ) y <-> E. z ( x ( B |` C ) z /\ z A y ) )
17	14, 15, 16	3bitr4i 211	. 2 |- ( x ( ( A o. B ) |` C ) y <-> x ( A o. ( B |` C ) ) y )
18	1, 2, 17	eqbrriv 4699	1 |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   class class class wbr 3982   |` cres 4606   o. ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-co 4613  df-res 4616

This theorem is referenced by:  cocnvcnv2  5115  coires1  5121  relcoi1  5135  dftpos2  6229